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Applying the same procedure to any positive integer <math>n\!</math> produces an expression called the ''doubly recursive factorization'' of <math>n.~\!</math> Let <math>\mathbb{M}</math> be the set of positive integers and let <math>\mathcal{L}_\text{DRF}</math> be the set of possible DRF expressions.  Then the procedure just illustrated defines a mapping <math>\operatorname{drf} : \mathbb{M} \to \mathcal{L}_\text{DRF}</math> and the doubly recursive factorization of <math>n\!</math> is denotable as <math>\operatorname{drf}(n).\!</math>

Applying the same procedure to any positive integer <math>n\!</math> produces an expression called the ''doubly recursive factorization'' of <math>n.\!</math> &nbsp; Let <math>\mathbb{M}</math> be the set of positive integers and let <math>\mathcal{L}</math> be the set of doubly recursive factorization expressions.  Then the procedure just illustrated defines a mapping <math>\operatorname{drf} : \mathbb{M} \to \mathcal{L}</math> that allows the doubly recursive factorization of <math>n\!</math> to be denoted <math>\operatorname{drf}(n).\!</math>

The forms of DRF expressions can be mapped into either one of two classes of graph-theoretical structures, called ''riffs'' and ''rotes'', respectively.

The forms of DRF expressions can be mapped into either one of two classes of graph-theoretical structures, called ''riffs'' and ''rotes'', respectively.